Simulation is an indispensable verification step before committing an integrated circuit to an expensive manufacturing process. An important step in circuit simulation is model evaluation. Modern analytical models such as bsim3 and bsim4 have become more and more complicated and expensive to evaluate in simulation. The percentage of model evaluation time in circuit simulation can be as high as 70–80%. This number can grow to 90% if one includes simulation time spent in using all the individual device evaluations to generate the circuit equations. As a result, model evaluation has become a bottle neck in the improvement of circuit simulation efficiency.
It is becoming more and more difficult to maintain some degree of smoothness in the approximated analytical model due to its complexity. As a result, many resources have been spent in checking the smoothness of device models and the consistency between function values and their derivatives in both the Electronic Design Automation (EDA) and the semi-conductor industries.
A high-order accurate table model is a potential solution to the above problems. Table models provide polynomial approximations of analytical models based on table-grid interpolations of the original current-voltage and charge-voltage (i-v/q-v) curves. Most existing table models use spline interpolation based on a fixed stencil consisting of a fixed group of consecutive interpolation points. Fix stencil interpolations work well if the curve to be approximated is globally smooth. The problem associated with fixed stencil interpolations, however, is that for curves containing a C0 corner point (i.e. curves that are continuous but not differentiable at the comer point), fixed stencil interpolation of second or higher order accuracy will be necessarily oscillatory near the C0 comer point.
The impact of this oscillatory behavior can be illustrated using the example in FIG. 1. Shown in FIG. 1 is the 3rd-order polynomial approximation with a fixed stencil for the step function. Notice the obvious undershoots near the bottom corner point. Such oscillations are denoted as Gibbs Phenomena in spectral methods. The main characteristics of these oscillations is that they do not decay in magnitude when the mesh or table grid is refined. In other words, no matter how big the table size is (or how large the number of interpolations is), the oscillations still remain at the same magnitude. Besides causing inaccuracy, the non-diminishing oscillations may cause the Newton iteration method to fail in solving nonlinear circuit equations.
Besides causing non-diminishing oscillations, existing table model-based simulation methods also lead to unnecessary memory usage and inefficient memory access. This is because these methods construct and store all the measurement data entries (i. e. table entries) at the beginning of simulation. Storing all the data entries is not necessary since simulations usually only concentrates on “local” data entries surrounding one or more operating points. Moreover, these data entries may contain redundant information since boundaries of these data entries may overlap. Storing these redundant information is certainly unnecessary. Moreover, storing all the table entries at the beginning of simulation makes it impractical to build the table entries adaptively and accurately due to the huge memory consumption.
There is therefore a need to develop a method and system for obtaining smooth, accurate, and computationally efficient approximation of analytical device models.